## Abstract

A family of sets ℱ is said to satisfy the (p, q) property if among any p sets in ℱ some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p ≥ q ≥ d + 1 there exists a minimum integer c = HD_{d}(p, q), such that any finite family of convex sets in ℝ^{d} that satisfies the (p, q) property can be pierced by at most c points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on HD_{d}(p, q), known as ‘the Hadwiger-Debrunner numbers’, is still a major open problem in discrete and computational geometry. The best known upper bound on the Hadwiger-Debrunner numbers in the plane is O(p(1.5+δ)(1+1q−2)) (for any δ > 0 and p ≥ q ≥ q_{0}(δ)), obtained by combining the results of Keller, Smorodinsky and Tardos (2017) and of Rubin (2018). The best lower bound is HD2(p,q)=Ω(pqlog(pq)), obtained by Bukh, Matoušek and Nivasch more than 10 years ago. In this paper we improve the lower bound significantly by showing that HD_{2}(p, q) ≥ p^{1+Ω(1/q)}. Furthermore, the bound is obtained by a family of lines and is tight for all families that have a bounded VC-dimension. Unlike previous bounds on the Hadwiger-Debrunner numbers, which mainly used the weak epsilon-net theorem, our bound stems from a surprising connection of the (p, q) problem to an old problem of Erdős on points in general position in the plane. We use a novel construction for Erdős’ problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on HD_{2}(p, 3). We then generalize the bound to HD_{2}(p, q) for q ≥ 3.

Original language | English |
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Pages (from-to) | 649-680 |

Number of pages | 32 |

Journal | Israel Journal of Mathematics |

Volume | 244 |

Issue number | 2 |

DOIs | |

State | Published - 1 Sep 2021 |

## ASJC Scopus subject areas

- General Mathematics